Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter...
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![Page 1: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/1.jpg)
Mike Paterson
Overhang bounds
Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup
and Peter Winkler
![Page 2: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/2.jpg)
![Page 3: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/3.jpg)
The classical solution
Harmonic Stacks
Using n blocks we can get an overhang of
1
2+
14
+16
+L +12n
=
1
21+
1
2+
1
3+L +
1
n⎛⎝⎜
⎞⎠⎟
≈1
2loge n
![Page 4: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/4.jpg)
Is the classical solution optimal?
Obviously not!
![Page 5: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/5.jpg)
Inverted triangles?
Balanced?
![Page 6: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/6.jpg)
???
![Page 7: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/7.jpg)
Inverted triangles?
Balanced?
![Page 8: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/8.jpg)
Inverted triangles?
Unbalanced!
![Page 9: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/9.jpg)
Inverted triangles?
Unbalanced!
![Page 10: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/10.jpg)
Diamonds?
Balanced?
![Page 11: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/11.jpg)
Diamonds?
The 4-diamond is balanced
![Page 12: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/12.jpg)
Diamonds?
The 5-diamond is …
![Page 13: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/13.jpg)
Diamonds?
… unbalanced!
![Page 14: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/14.jpg)
What really happens?
![Page 15: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/15.jpg)
What really happens!
![Page 16: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/16.jpg)
How do we know this is unbalanced?
![Page 17: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/17.jpg)
… and this balanced?
![Page 18: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/18.jpg)
Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
![Page 19: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/19.jpg)
Checking balance
![Page 20: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/20.jpg)
Checking balance
F1F2 F3 F4 F5 F6
F7F8 F9 F10
F11 F12
F13F14 F15 F16
F17 F18
Equivalent to the feasibilityof a set of linear inequalities:
![Page 21: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/21.jpg)
Small optimal stacks
Overhang = 1.16789Blocks = 4
Overhang = 1.30455Blocks = 5
Overhang = 1.4367Blocks = 6
Overhang = 1.53005Blocks = 7
![Page 22: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/22.jpg)
Small optimal stacks
Overhang = 2.14384Blocks = 16
Overhang = 2.1909Blocks = 17
Overhang = 2.23457Blocks = 18
Overhang = 2.27713Blocks = 19
![Page 23: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/23.jpg)
Support and balancing blocks
Principalblock
Support set
Balancing
set
![Page 24: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/24.jpg)
Support and balancing blocks
Principalblock
Support set
Balancing
set
![Page 25: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/25.jpg)
Principalblock
Support set
Stacks with downward external
forces acting on them
Loaded stacks
Size =
number of blocks
+ sum of external
forces
![Page 26: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/26.jpg)
Principalblock
Support set
Stacks in which the support set contains
only one block at each level
Spinal stacks
![Page 27: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/27.jpg)
Optimal spinal stacks
…
Optimality condition:
![Page 28: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/28.jpg)
Spinal overhang
Let S (n) be the maximal overhang achievable using a spinal stack with n blocks.
Let S*(n) be the maximal overhang achievable using a loaded spinal stack on total weight n.
Theorem:
A factor of 2 improvement over harmonic stacks!
Conjecture:
![Page 29: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/29.jpg)
Optimal weight 100 loaded spinal stack
![Page 30: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/30.jpg)
Optimal 100-block spinal stack
![Page 31: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/31.jpg)
Are spinal stacks optimal?
No!
Support set is not spinal!
Overhang = 2.32014Blocks = 20
Tiny gap
![Page 32: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/32.jpg)
Optimal 30-block stack
Overhang = 2.70909Blocks = 30
![Page 33: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/33.jpg)
Optimal (?) weight 100 construction
Overhang = 4.2390Blocks = 49
Weight = 100
![Page 34: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/34.jpg)
“Parabolic” constructions
6-stack
Number of blocks: Overhang:
Balanced!
![Page 35: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/35.jpg)
“Parabolic” constructions
6-slab
5-slab
4-slab
![Page 36: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/36.jpg)
r-slab
r-slab
![Page 37: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/37.jpg)
r-slab within an (r +1)-slab
![Page 38: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/38.jpg)
![Page 39: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/39.jpg)
An exponential improvement over the ln n overhang of spinal stacks !!!
So with n blocks we can
get an overhang of c n1/3
for some constant c !!!
Note: c n1/3 ~ e1/3 ln n
Overhang, Paterson & Zwick, American Math. Monthly Jan 2009
![Page 40: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/40.jpg)
What is really the best design?
Some experimental results with optimised “brick-wall”
constructions
Firstly, symmetric designs
![Page 41: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/41.jpg)
“Vases”
Weight = 1151.76
Blocks = 1043
Overhang = 10
![Page 42: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/42.jpg)
“Vases”
Weight = 115467.
Blocks = 112421
Overhang = 50
![Page 43: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/43.jpg)
then, asymmetric designs
![Page 44: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/44.jpg)
“Oil lamps”
Weight = 1112.84
Blocks = 921
Overhang = 10
![Page 45: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/45.jpg)
Ωn1is a lower bound
for overhang with n blocks?
Can we do better? Not much!
Theorem: Maximum overhang is less than C n1/3 for some constant C
Maximum overhang, Paterson, Perez, Thorup, Winkler, Zwick, American Math. Monthly, Nov 2009
![Page 46: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/46.jpg)
Forces between blocks
Assumption: No friction.All forces are vertical.
Equivalent sets of forces
![Page 47: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/47.jpg)
![Page 48: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/48.jpg)
![Page 49: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/49.jpg)
![Page 50: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/50.jpg)
![Page 51: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/51.jpg)
![Page 52: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/52.jpg)
![Page 53: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/53.jpg)
Distributions
![Page 54: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/54.jpg)
Moments and spread
j-th moment
Center of mass
Spread
NB important measure
![Page 55: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/55.jpg)
Signed distributions
![Page 56: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/56.jpg)
MovesA move is a signed distribution with M0[ ] = M1[ ] =
0 whose support is contained in an interval of length 1
A move is applied by adding it to a distribution.
A move can be applied only if the resulting signed distribution is a distribution.
![Page 57: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/57.jpg)
Equilibrium
F1 + F2 + F3 = F4 + F5
x1 F1+ x2 F2+ x3 F3 = x4 F4+ x5 F5
Force equation
Moment equation
F1
F5F4
F3
F2
Recall!
![Page 58: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/58.jpg)
MovesA move is a signed distribution
with M0[ ] = M1[ ] = 0 whose support
is contained in an interval of length 1
A move is applied by adding it to a distribution.
A move can be applied only if the resulting signed distribution is a distribution.
![Page 59: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/59.jpg)
Move sequences
![Page 60: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/60.jpg)
Extreme moves
Moves all the mass within the interval to the endpoints
![Page 61: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/61.jpg)
Lossy moves
If is a move in [c-½,c+½] then
A lossy move removes one unit of mass from position c
Alternatively, a lossy move freezes one unit of mass at position c
![Page 62: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/62.jpg)
Overhang and mass movementIf there is an n-block stack that achieves an overhang of d, then
n–1 lossy moves
![Page 63: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/63.jpg)
Main theorem
![Page 64: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/64.jpg)
Four stepsShift half mass outside interval Shift half mass across interval
Shift some mass across intervaland no further
Shift some mass across interval
![Page 65: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/65.jpg)
Simplified setting
“Integral” distributions
Splitting moves
![Page 66: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/66.jpg)
0 1 2 3-3 -2 -1
![Page 67: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/67.jpg)
Basic challenge
Suppose that we start with a mass of 1 at the origin.How many splits are needed to get, say, half of the mass to distance d ?
Reminiscent of a random walk on the line
O(d3) splits are “clearly” sufficient
To prove: (d3) splits are required
![Page 68: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/68.jpg)
Effect of a split
Note that such split moves here have associated interval of length 2.
![Page 69: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/69.jpg)
Spread vs. second moment argument
![Page 70: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/70.jpg)
That’s a start!
we have to extend the proof to the general case, with general distributions and moves;
we need to get improved bounds for small values of p;
we have to show that moves beyond position d cannot help;
But …
we did not yet use the lossy nature of moves.
That’s another talk!
![Page 71: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/71.jpg)
Open problems
What is the asymptotic shape of “vases”? What is the asymptotic shape of “oil lamps”? What is the gap between brick-wall stacks
and general stacks? Other games! “Bridges” and “seesaws”.
![Page 72: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/72.jpg)
Design the best bridge
![Page 73: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/73.jpg)
Design the best seesaw
![Page 74: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/74.jpg)
A big open area
We only consider frictionless 2D constructions here. This implies no horizontal forces, so, even if blocks are tilted, our results still hold. What happens in the frictionless 3D case?
With friction, everything changes!
![Page 75: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/75.jpg)
With friction
With enough friction we can get overhang greater than 1 with only 2 blocks!
With enough friction, all diamonds are balanced, so we get Ω(n1/2) overhang.
Probably we can get Ω(n1/2) overhang with arbitrarily small friction.
With enough friction, there are possibilities to get exponents greater than 1/2.
In 3D, I think that when the coefficient of friction is greater than 1 we can get Ω(n) overhang.
![Page 76: Mike Paterson Overhang bounds Joint work with Uri Zwick, Yuval Peres, Mikkel Thorup and Peter Winkler.](https://reader036.fdocuments.net/reader036/viewer/2022062518/56649f585503460f94c7d181/html5/thumbnails/76.jpg)
The end
Applications?